Fermi surface nesting and CDW/SDW/SC orders.
What is the definition of a nesting vector?
And why Fermi surface nesting gives rise to different orders at $T=0$?
(CDW: charge density wave; SDW: spin density wave; SC: superconductivity)
Answer
A nesting vector is a single vector in momentum space that connects large, parallel regions of a fermi surface. For example, a square lattice at half-filling has a fermi surface that looks like a diamond. If you have some scattering mechanism that takes an electron and adds a momentum $\vec{Q}$ to it, like a hamiltonian term $$g(\vec{k},\vec{Q}) \cdot c_{k+Q}^{\dagger}c_{k}^{\phantom{\dagger}}$$ then you can easily see that there's a single vector $\vec{Q}=\frac{1}{a}(\pi,\pi)$ that connects a whole lot of states in the Brillouin zone. In perturbation theory we get terms that go like $\frac{1}{E_k - E_{k+Q}}$, so if a single vector $Q$ connects lots of $k$ states (which have the same energy, by virtue of making up the fermi surface, and the states right around the fermi surface are the only ones that matter for low temperature stuff like CDW/SDW) then the static structure factor is going to be strongly peaked at that $Q$.
Fermi surface nesting gives rise to different orders because the scattering mechanisms of different materials are different. The term I wrote above didn't have any spin index on the operators. Spin preserving scattering like that would lead to a CDW. But if I had magnetic scattering that flipped the spin of the electron, I would have had an SDW state.
Lastly, superconductivity is a bit different. You don't need any nesting at all to get SC since it joins electron pairs of opposite spin and quasimomentum (or more generally time reversed pairs). This is why superconductivity is a much more robust phenomena than density waves.
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